numpy似乎为循环矩阵返回了错误的特征向量

Question

我用numpy来计算循环矩阵的特征值和特征向量。以下是我的代码（j=1,2…6的Hji是预定义的）：

>>> import numpy as np
>>> H = np.array([H1i, H2i, H3i, H4i, H5i, H6i])
>>> H
array([[ 0.,  1.,  0.,  0.,  0.,  1.],
       [ 1.,  0.,  1.,  0.,  0.,  0.],
       [ 0.,  1.,  0.,  1.,  0.,  0.],
       [ 0.,  0.,  1.,  0.,  1.,  0.],
       [ 0.,  0.,  0.,  1.,  0.,  1.],
       [ 1.,  0.,  0.,  0.,  1.,  0.]])
>>> from numpy import linalg as LA
>>> w, v = LA.eig(H)

>>> w
array([-2.,  2.,  1., -1., -1.,  1.])
>>> v
array([[ 0.40824829, -0.40824829, -0.57735027,  0.57732307,  0.06604706,
         0.09791921],
       [-0.40824829, -0.40824829, -0.28867513, -0.29351503, -0.5297411 ,
        -0.4437968 ],
       [ 0.40824829, -0.40824829,  0.28867513, -0.28380804,  0.46369403,
        -0.54171601],
       [-0.40824829, -0.40824829,  0.57735027,  0.57732307,  0.06604706,
        -0.09791921],
       [ 0.40824829, -0.40824829,  0.28867513, -0.29351503, -0.5297411 ,
         0.4437968 ],
       [-0.40824829, -0.40824829, -0.28867513, -0.28380804,  0.46369403,
         0.54171601]])

特征值是正确的。但是，对于特征向量，我发现它们不是线性无关的

^{pr2}$

你可以看到有非对角项（检查V[2,5]=-1.69601044e-01），这意味着它们不是线性独立向量。既然这是一个厄米矩阵，它的特征向量如何变得相互依赖？在

顺便说一下，我也用matlab来计算它，它返回正确的值

V =

    0.4082   -0.2887   -0.5000    0.5000    0.2887   -0.4082
   -0.4082   -0.2887    0.5000    0.5000   -0.2887   -0.4082
    0.4082    0.5774         0         0   -0.5774   -0.4082
   -0.4082   -0.2887   -0.5000   -0.5000   -0.2887   -0.4082
    0.4082   -0.2887    0.5000   -0.5000    0.2887   -0.4082
   -0.4082    0.5774         0         0    0.5774   -0.4082


D =

   -2.0000         0         0         0         0         0
         0   -1.0000         0         0         0         0
         0         0   -1.0000         0         0         0
         0         0         0    1.0000         0         0
         0         0         0         0    1.0000         0
         0         0         0         0         0    2.0000

Answer 1

对于厄米矩阵和对称矩阵，您应该使用另一个函数：eigh。在

import numpy as np
from numpy import linalg as LA

H = np.array([[ 0.,  1.,  0.,  0.,  0.,  1.],
       [ 1.,  0.,  1.,  0.,  0.,  0.],
       [ 0.,  1.,  0.,  1.,  0.,  0.],
       [ 0.,  0.,  1.,  0.,  1.,  0.],
       [ 0.,  0.,  0.,  1.,  0.,  1.],
       [ 1.,  0.,  0.,  0.,  1.,  0.]])

w, v = LA.eigh(H)

V = np.zeros((6,6))
for i in range(6):
    for j in range(6):
        V[i,j] = np.dot(v[:,i], v[:,j])

w
Out[19]: array([-2., -1., -1.,  1.,  1.,  2.])

v
Out[20]: 
array([[-0.40824829, -0.57735027,  0.        ,  0.        ,  0.57735027,
         0.40824829],
       [ 0.40824829,  0.28867513, -0.5       , -0.5       ,  0.28867513,
         0.40824829],
       [-0.40824829,  0.28867513,  0.5       , -0.5       , -0.28867513,
         0.40824829],
       [ 0.40824829, -0.57735027,  0.        ,  0.        , -0.57735027,
         0.40824829],
       [-0.40824829,  0.28867513, -0.5       ,  0.5       , -0.28867513,
         0.40824829],
       [ 0.40824829,  0.28867513,  0.5       ,  0.5       ,  0.28867513,
         0.40824829]])

V
Out[21]: 
array([[  1.00000000e+00,   8.32667268e-17,   2.77555756e-17,
          8.32667268e-17,  -2.08166817e-16,   0.00000000e+00],
       [  8.32667268e-17,   1.00000000e+00,   5.55111512e-17,
          5.55111512e-17,  -2.22044605e-16,  -1.11022302e-16],
       [  2.77555756e-17,   5.55111512e-17,   1.00000000e+00,
          0.00000000e+00,   2.77555756e-17,   1.11022302e-16],
       [  8.32667268e-17,   5.55111512e-17,   0.00000000e+00,
          1.00000000e+00,   8.32667268e-17,   5.55111512e-17],
       [ -2.08166817e-16,  -2.22044605e-16,   2.77555756e-17,
          8.32667268e-17,   1.00000000e+00,   0.00000000e+00],
       [  0.00000000e+00,  -1.11022302e-16,   1.11022302e-16,
          5.55111512e-17,   0.00000000e+00,   1.00000000e+00]])